I figured it out.  It was a careless mistake of not setting the $img->owner_guid to the $group->owner_guid.  I had it set to $group->guid on accident.

I am still curious as to why $img->delete() returns 'false' even when it does successfully delete the icons for that group though (I have it in my owner php script so that I can call it by using jQuery).

<?php
class            Achievement
        extends    ElggGroup
{
    ////////////////////////////////////////
    function __contruct($guid = null)
    {
        parent::__construct($guid);
    }//function __contruct($guid = null)
    ////////////////////////////////////////
    function leave($user, $force_stay = false)
    {
        if ($force_stay)
        {
            display_message('You Cannot Leave an Achievement.');
            return false;
        }
        else
        {
            display_message('OK You Left an Achievement.');
            parent::leave($user);// Where does this logic go to ???
        }//if ($force_stay)
    }//function leave($user, $force_stay = false)
    ////////////////////////////////////////
}
?>

Ok, so I must have been out of it when I wrote this... geez. Anyways, it still doesn't work.  I have put a die('crap'); statement at the beginning of the function and it doesn't get called...  So, my leave function must not even be called, right? 

As for this:

parent::leave($user);// Where does this logic go to ???

This calls the original leave(ElggUser) function defined in the Elgg core system.  Since $force_stay is false, the user is allowed to leave the group. 

I just noticed that the name of $force_stay should be $can_leave.  I want to have the original leave() function work fine, so $achv->leave($user) works like before, but if you do $achv->leave($user, true), the user cannot leave the group.  Here is exactly what I have in that file now:

<?php

// Achievement class

class Achievement extends ElggGroup {

        function __contruct($guid = null) {
                parent::__construct($guid);
        }

        function leave($user, $can_leave = false) {

                die('crap');

                if ($can_leave) {
                        echo 'The user is leaving the group';
                        parent::leave($user);
                } else {
                        echo 'The user is forced to stay in the group';
                        return false;
                        //display_message('You cannot leave an achievement.');
                }
                die('end');
        }
}

Do I need to declare Achievement as a new enitity type as well in my start.php?

It saves it as metadata. Have you had a look at the data model discussion in the docs? Especially the part about metadata.

Don't forget to call $entity->save() after to persist the data in the db.

Thanks Brett! I completely missed that second page apparently!  Sorry for the nub question.

Np. The interface for this is a bit confusing because it requires knowledge of the data model.